Vector systems of Painlev\'e type

TL;DR

This paper develops vector generalizations of classical integrable equations, resulting in new Painlevé-type systems with isomonodromic Lax representations and connections to known integrable models.

Contribution

It introduces vector systems of Painlevé type derived from reductions of NLS, mKdV, and KdV equations, expanding the landscape of integrable and Painlevé equations.

Findings

Derived vector Painlevé systems admit isomonodromic Lax representations.

Some systems are nonautonomous deformations of classical integrable models.

Established a connection with quasiperiodic dressing chain equations.

Abstract

The group reduction procedure is applied to vector generalizations of the NLS, mKdV, and KdV equations. The resulting ODE systems admit isomonodromic Lax representations and are multicomponent generalizations of the Painlev\'e equations P$_1$, P$_2$, P$_{34}$, and P$_4$. Some of them can be interpreted as nonautonomous deformations of well-known systems integrable in the Liouville sense, in particular, the Garnier and H\'enon--Heiles systems. In one case, an unexpected connection with the equations of quasiperiodic dressing chain for the Schr\"odinger operator is established.